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Your data matches 49 different statistics following compositions of up to 3 maps.
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Matching statistic: St000223
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(load all 3 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,2,4,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,4,1,2] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,3,5,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,4,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,5,2,3] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,2,3,5,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,2,4,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,2,5,4,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,2,4,5,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,2,5,1,3] => 1
Description
The number of nestings in the permutation.
Matching statistic: St000371
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(load all 3 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,2,4,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,4,1,2] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,3,5,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,4,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,5,2,3] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,2,3,5,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,2,4,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,2,5,4,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,2,4,5,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,2,5,1,3] => 1
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation π of {1,…,n}, this is the number of indices j such that there exist indices i,k with i<j<k and π(i)>π(j)>π(k). In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St000373
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,2,4,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,4,1,2] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,3,5,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,4,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,5,2,3] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,2,3,5,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,2,4,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,2,5,4,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,2,4,5,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,2,5,1,3] => 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => ? = 0
Description
The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length 3.
Given a permutation π=[π1,…,πn], this statistic counts the number of position j such that πj≥j and there exist indices i,k with i<j<k and πi>πj>πk.
See also [[St000213]] and [[St000119]].
Matching statistic: St000123
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [3,2,1] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => [3,1,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => [1,4,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,2,4,1] => [4,2,1,3] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => [3,1,2,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => [4,1,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => [4,1,2,3] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,4,1,2] => [3,4,1,2] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,3,5,2] => [1,5,3,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,4,2] => [1,5,2,4,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => [1,5,2,3,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => [2,1,5,3,4] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,2,3,5,1] => [5,2,3,1,4] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,2,4,1,5] => [4,2,1,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,2,5,4,1] => [5,2,1,4,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,2,4,5,1] => [5,2,1,3,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,2,5,1,3] => [4,2,5,1,3] => 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,3,5,4,7,6,8,2] => [1,4,3,6,5,8,7,2] => [1,8,3,2,5,4,7,6] => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,3,6,8,7,5] => [2,1,4,3,7,6,8,5] => [2,1,4,3,8,6,5,7] => ? = 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,7,6,8,5] => [2,1,4,3,6,8,7,5] => [2,1,4,3,8,5,7,6] => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,1,4,6,5,3,8,7] => [2,1,5,4,6,3,8,7] => [2,1,6,4,3,5,8,7] => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,1,4,6,5,8,7,3] => [2,1,5,4,7,6,8,3] => [2,1,8,4,3,6,5,7] => ? = 2
[1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,1,4,8,6,5,7,3] => [2,1,6,4,7,3,8,5] => [2,1,6,4,8,3,5,7] => ? = 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,4,6,3,8,7] => [2,1,4,6,5,3,8,7] => [2,1,6,3,5,4,8,7] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,1,6,4,5,7,8,3] => [2,1,4,5,8,6,7,3] => [2,1,8,3,4,6,7,5] => ? = 2
[1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,7,6,8,4] => [3,2,1,6,5,8,7,4] => [3,2,1,8,5,4,7,6] => ? = 3
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,4,3,1,6,5,8,7] => [3,2,4,1,6,5,8,7] => [4,2,1,3,6,5,8,7] => ? = 1
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,4,3,1,6,8,7,5] => [3,2,4,1,7,6,8,5] => [4,2,1,3,8,6,5,7] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [2,4,3,5,1,7,8,6] => [3,2,5,4,1,8,7,6] => [5,2,1,4,3,8,7,6] => ? = 3
[1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [2,4,3,6,5,1,8,7] => [3,2,5,4,6,1,8,7] => [6,2,1,4,3,5,8,7] => ? = 2
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [2,4,3,6,5,7,1,8] => [3,2,5,4,7,6,1,8] => [7,2,1,4,3,6,5,8] => ? = 3
[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,8,7,1] => [3,2,5,4,7,6,8,1] => [8,2,1,4,3,6,5,7] => ? = 3
[1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [2,6,4,3,5,1,8,7] => [4,2,5,1,6,3,8,7] => [4,2,6,1,3,5,8,7] => ? = 1
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [3,2,4,1,6,5,8,7] => [2,4,3,1,6,5,8,7] => [4,1,3,2,6,5,8,7] => ? = 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [3,2,4,1,7,6,8,5] => [2,4,3,1,6,8,7,5] => [4,1,3,2,8,5,7,6] => ? = 2
[1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> [4,2,3,5,6,1,8,7] => [2,3,6,4,5,1,8,7] => [6,1,2,4,5,3,8,7] => ? = 2
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [5,2,3,4,6,7,8,1] => [2,3,4,8,5,6,7,1] => [8,1,2,3,5,6,7,4] => ? = 3
[1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [8,2,4,3,6,5,7,1] => [2,4,6,1,7,3,8,5] => [4,1,6,2,8,3,5,7] => ? = 0
Description
The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map.
The Simion-Schmidt map takes a permutation and turns each occcurrence of [3,2,1] into an occurrence of [3,1,2], thus reducing the number of inversions of the permutation. This statistic records the difference in length of the permutation and its image.
Apparently, this statistic can be described as the number of occurrences of the mesh pattern ([3,2,1], {(0,3),(0,2)}). Equivalent mesh patterns are ([3,2,1], {(0,2),(1,2)}), ([3,2,1], {(0,3),(1,3)}) and ([3,2,1], {(1,2),(1,3)}).
Matching statistic: St000731
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => ? = 2
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,4,5,1,3,6,7] => ? = 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,5,6,7,1,3,4] => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,5,2,7,4,8,6] => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,8,5,7] => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,3,5,8,7] => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,4,6,3,8,5,7] => ? = 2
[1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,4,6,8,3,5,7] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7,8] => ? = 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1,5,7,4,8,6] => ? = 3
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,8,1,5,6,7] => ? = 3
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,8,1,4,5,6,7] => ? = 2
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [2,4,1,3,6,5,8,7] => ? = 1
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [2,4,1,3,6,8,5,7] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,5,3,7,8,6] => ? = 3
[1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [2,4,1,6,3,5,8,7] => ? = 2
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5,8] => ? = 3
[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,1,6,3,8,5,7] => ? = 3
[1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4,5,1,3,6,7,8] => ? = 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,4,6,1,3,5,8,7] => ? = 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [2,5,6,7,1,3,4,8] => ? = 1
[1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0,1,0]
=> [4,1,2,5,6,3,8,7] => ? = 2
[1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,1,0,0]
=> [4,1,2,5,6,8,3,7] => ? = 3
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [5,1,2,3,6,7,8,4] => ? = 3
[1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,1,6,2,8,3,5,7] => ? = 0
[1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [3,5,7,1,8,2,4,6] => ? = 2
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,9,1,4,5,6,7,8] => ? = 2
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [2,6,7,8,9,1,3,4,5] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7,8,9] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7,8,9,10] => ? = 1
Description
The number of double exceedences of a permutation.
A double exceedence is an index σ(i) such that i<σ(i)<σ(σ(i)).
Matching statistic: St000366
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,3,2,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,2,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,3,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,4,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,3,2,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,2,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,4,2,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,4,5,2,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,2,1] => 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [1,5,4,6,3,7,2] => ? = 0
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,7,6,1] => [6,7,5,4,3,2,1] => ? = 4
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,7,5,6,1] => [5,6,7,4,3,2,1] => ? = 3
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,7,4,5,6,1] => [4,5,6,7,3,2,1] => ? = 2
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,7,3,4,5,6,1] => [3,4,5,6,7,2,1] => ? = 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,5,4,3,1,6,7] => [4,3,5,2,1,6,7] => ? = 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => [5,4,6,3,7,2,1] => ? = 1
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [7,2,3,4,6,5,1] => [2,3,4,6,5,7,1] => ? = 0
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [7,2,3,6,5,4,1] => [2,3,5,6,4,7,1] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => [3,2,4,1,5,6,7] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [4,3,5,2,6,1,7] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,6,5,4,3,7,8] => [1,2,5,4,6,3,7,8] => ? = 0
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,3,5,4,7,6,8,2] => [1,4,6,8,7,5,3,2] => ? = 3
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,3,6,5,4,7,2,8] => [1,5,4,7,6,3,2,8] => ? = 2
[1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,7,3,5,4,6,2,8] => [1,3,5,4,6,7,2,8] => ? = 0
[1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,7,6,4,5,3,2,8] => [1,4,5,6,3,7,2,8] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,3,6,8,7,5] => [2,1,4,3,7,8,6,5] => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,1,4,6,5,3,8,7] => [2,1,5,6,4,3,8,7] => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,1,4,6,5,8,7,3] => [2,1,5,7,8,6,4,3] => ? = 2
[1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,1,4,8,6,5,7,3] => [2,1,6,5,7,8,4,3] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,1,6,4,5,7,8,3] => [2,1,4,5,8,7,6,3] => ? = 2
[1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,7,6,8,4] => [3,2,1,6,8,7,5,4] => ? = 3
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,4,8,5,6,7,1] => [5,6,7,8,4,3,2,1] => ? = 3
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,3,8,4,5,6,7,1] => [4,5,6,7,8,3,2,1] => ? = 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,6,5,4,7,8,1] => [5,4,8,7,6,3,2,1] => ? = 4
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,4,3,1,6,5,8,7] => [3,4,2,1,6,5,8,7] => ? = 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [2,4,3,1,7,6,8,5] => [3,4,2,1,6,8,7,5] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [2,4,3,5,1,7,8,6] => [3,5,4,2,1,8,7,6] => ? = 3
[1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [2,4,3,6,5,1,8,7] => [3,5,6,4,2,1,8,7] => ? = 2
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [2,4,3,6,5,7,1,8] => [3,5,7,6,4,2,1,8] => ? = 3
[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,8,7,1] => [3,5,7,8,6,4,2,1] => ? = 3
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,8,3,4,5,6,7,1] => [3,4,5,6,7,8,2,1] => ? = 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [2,7,3,5,4,6,8,1] => [3,5,4,6,8,7,2,1] => ? = 2
[1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,5,4,3,1,6,7,8] => [4,3,5,2,1,6,7,8] => ? = 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [2,6,4,3,5,1,8,7] => [4,3,5,6,2,1,8,7] => ? = 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [2,7,6,4,5,3,8,1] => [4,5,6,3,8,7,2,1] => ? = 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,7,6,5,4,3,1,8] => [5,4,6,3,7,2,1,8] => ? = 1
[1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [3,2,4,1,6,8,7,5] => [2,4,3,1,7,8,6,5] => ? = 2
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [3,2,6,5,4,8,7,1] => [2,5,4,7,8,6,3,1] => ? = 2
[1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> [4,2,3,5,6,1,8,7] => [2,3,6,5,4,1,8,7] => ? = 2
[1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
=> [4,2,3,5,6,8,7,1] => [2,3,7,8,6,5,4,1] => ? = 3
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [5,2,3,4,6,7,8,1] => [2,3,4,8,7,6,5,1] => ? = 3
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [8,2,3,4,5,7,6,1] => [2,3,4,5,7,6,8,1] => ? = 0
[1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> [8,2,3,5,4,6,7,1] => [2,3,5,4,6,7,8,1] => ? = 0
[1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [8,2,4,3,6,5,7,1] => [2,4,3,6,5,7,8,1] => ? = 0
[1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> [8,2,6,4,5,3,7,1] => [2,4,5,6,3,7,8,1] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8] => [3,2,4,1,5,6,7,8] => ? = 0
[1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [8,3,5,4,7,6,2,1] => [4,6,7,5,3,2,8,1] => ? = 2
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [8,7,3,4,5,6,2,1] => [3,4,5,6,7,2,8,1] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8] => [4,3,5,2,6,1,7,8] => ? = 0
Description
The number of double descents of a permutation.
A double descent of a permutation π is a position i such that π(i)>π(i+1)>π(i+2).
Matching statistic: St000931
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1
[1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 3
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 4
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 3
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 1
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 2
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 2
[1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 3
[1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 2
[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 3
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2
[1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 2
[1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> ? = 3
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 3
[1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 0
[1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 0
[1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 0
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 0
[1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 0
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 0
[1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 0
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
Description
The number of occurrences of the pattern UUU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St000356
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [3,1,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => [2,1,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [2,4,3,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [5,2,4,3,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [5,1,4,3,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,2,3] => [5,2,1,4,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,2,4] => [5,3,1,4,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => [5,3,2,1,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => [5,2,3,1,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [4,5,3,2,1] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [4,5,2,3,1] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [4,5,1,3,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => [4,5,2,1,3] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,5,4,1,2] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [2,5,4,3,1] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,1,2,3] => [2,1,5,4,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,1,2,5] => [3,2,5,4,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,1,2,4] => [3,1,5,4,2] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,4,5,1,2] => [3,2,1,5,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,1,2] => [2,3,1,5,4] => 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [1,5,4,6,3,7,2] => [7,3,4,2,5,1,6] => ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [6,7,5,4,3,2,1] => ? = 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7] => [3,1,2,4,5,6,7] => [5,7,6,4,3,2,1] => ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,7,5,6,1] => [5,6,7,1,2,3,4] => [3,2,1,7,6,5,4] => ? = 3
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,7,4,5,6,1] => [4,5,6,7,1,2,3] => [4,3,2,1,7,6,5] => ? = 2
[1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,5,4,3,1,6,7] => [4,3,5,1,2,6,7] => [4,5,3,7,6,2,1] => ? = 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => [5,4,6,3,7,1,2] => [3,4,2,5,1,7,6] => ? = 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7] => [2,3,1,4,5,6,7] => [6,5,7,4,3,2,1] => ? = 0
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [4,2,3,1,5,6,7] => [2,3,4,1,5,6,7] => [6,5,4,7,3,2,1] => ? = 0
[1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [4,2,3,1,5,7,6] => [2,3,4,1,5,7,6] => [6,5,4,7,3,1,2] => ? = 0
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [5,2,3,4,1,6,7] => [2,3,4,5,1,6,7] => [6,5,4,3,7,2,1] => ? = 0
[1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [5,2,3,4,1,7,6] => [2,3,4,5,1,7,6] => [6,5,4,3,7,1,2] => ? = 0
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,2,3,4,5,1,7] => [2,3,4,5,6,1,7] => [6,5,4,3,2,7,1] => ? = 0
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [7,2,3,4,6,5,1] => [2,3,4,6,5,7,1] => [6,5,4,2,3,1,7] => ? = 0
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [7,2,3,6,5,4,1] => [2,3,5,6,4,7,1] => [6,5,3,2,4,1,7] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => [3,2,4,1,5,6,7] => [5,6,4,7,3,2,1] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [4,3,5,2,6,1,7] => [4,5,3,6,2,7,1] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,7,6,8] => [1,2,3,4,5,7,6,8] => [8,7,6,5,4,2,3,1] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,3,5,4,6,7,8] => [1,2,3,5,4,6,7,8] => [8,7,6,4,5,3,2,1] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,6,5,4,3,7,8] => [1,2,5,4,6,3,7,8] => [8,7,4,5,3,6,2,1] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8] => [1,3,2,4,5,6,7,8] => [8,6,7,5,4,3,2,1] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,7,6,8] => [1,3,2,5,4,7,6,8] => [8,6,7,4,5,2,3,1] => ? = 0
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,3,5,4,7,6,8,2] => [1,4,6,8,2,3,5,7] => [8,5,3,1,7,6,4,2] => ? = 3
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,3,6,5,4,7,2,8] => [1,5,4,7,2,3,6,8] => [8,4,5,2,7,6,3,1] => ? = 2
[1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,7,3,5,4,6,2,8] => [1,3,5,4,6,7,2,8] => [8,6,4,5,3,2,7,1] => ? = 0
[1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,7,6,4,5,3,2,8] => [1,4,5,6,3,7,2,8] => [8,5,4,3,6,2,7,1] => ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [7,8,6,5,4,3,2,1] => ? = 0
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,7,6,8,5] => [2,1,4,3,6,8,5,7] => [7,8,5,6,3,1,4,2] => ? = 1
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,8,7,6,5] => [2,1,4,3,7,6,8,5] => [7,8,5,6,2,3,1,4] => ? = 0
[1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,1,4,6,5,8,7,3] => [2,1,5,7,8,3,4,6] => [7,8,4,2,1,6,5,3] => ? = 2
[1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,1,4,8,6,5,7,3] => [2,1,6,5,7,8,3,4] => [7,8,3,4,2,1,6,5] => ? = 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,4,6,3,8,7] => [2,1,4,6,3,5,8,7] => [7,8,5,3,6,4,1,2] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,1,6,4,5,7,8,3] => [2,1,4,5,8,3,6,7] => [7,8,5,4,1,6,3,2] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,6,5,4,3,8,7] => [2,1,5,4,6,3,8,7] => [7,8,4,5,3,6,1,2] => ? = 0
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,1,8,5,4,7,6,3] => [2,1,5,4,7,6,8,3] => [7,8,4,5,2,3,1,6] => ? = 0
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => [7,8,3,4,2,5,1,6] => ? = 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8] => [3,1,2,4,5,6,7,8] => [6,8,7,5,4,3,2,1] => ? = 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,7,6,8,4] => [3,1,2,6,8,4,5,7] => [6,8,7,3,1,5,4,2] => ? = 3
[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,6,5,4,7,8,1] => [5,4,8,1,2,3,6,7] => [4,5,1,8,7,6,3,2] => ? = 4
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [2,4,3,1,7,6,8,5] => [3,4,1,2,6,8,5,7] => [6,5,8,7,3,1,4,2] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [2,4,3,5,1,7,8,6] => [3,5,1,2,4,8,6,7] => [6,4,8,7,5,1,3,2] => ? = 3
[1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [2,4,3,6,5,1,8,7] => [3,5,6,1,2,4,8,7] => [6,4,3,8,7,5,1,2] => ? = 2
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [2,4,3,6,5,7,1,8] => [3,5,7,1,2,4,6,8] => [6,4,2,8,7,5,3,1] => ? = 3
[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,8,7,1] => [3,5,7,8,1,2,4,6] => [6,4,2,1,8,7,5,3] => ? = 3
[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [2,7,3,5,4,6,8,1] => [3,5,4,6,8,1,2,7] => [6,4,5,3,1,8,7,2] => ? = 2
[1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,5,4,3,1,6,7,8] => [4,3,5,1,2,6,7,8] => [5,6,4,8,7,3,2,1] => ? = 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [2,6,4,3,5,1,8,7] => [4,3,5,6,1,2,8,7] => [5,6,4,3,8,7,1,2] => ? = 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [2,7,6,4,5,3,8,1] => [4,5,6,3,8,1,2,7] => [5,4,3,6,1,8,7,2] => ? = 2
Description
The number of occurrences of the pattern 13-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern 13−2.
Matching statistic: St001483
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
St001483: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 2 + 1
[1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 0 + 1
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 0 + 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 3 + 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4 + 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 4 + 1
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 2 + 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 2 + 1
[1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 + 1
[1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 2 + 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 3 + 1
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 1 + 1
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
Matching statistic: St001066
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 0 + 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 0 + 1
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 0 + 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 3 + 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 4 + 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 4 + 1
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 1 + 1
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 3 + 1
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 1 + 1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
The following 39 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000358The number of occurrences of the pattern 31-2. St000732The number of double deficiencies of a permutation. St000365The number of double ascents of a permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St000039The number of crossings of a permutation. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St000317The cycle descent number of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000237The number of small exceedances. St001061The number of indices that are both descents and recoils of a permutation. St000485The length of the longest cycle of a permutation. St000649The number of 3-excedences of a permutation. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000451The length of the longest pattern of the form k 1 2. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000058The order of a permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St001864The number of excedances of a signed permutation. St001964The interval resolution global dimension of a poset. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001550The number of inversions between exceedances where the greater exceedance is linked. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
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